Run Profiler against the secondary non-readable instance to catch attempted sessions and logins being made to the instance.

Are you doing this on your non-readable secondary database? If yes, then you are seeing the correct message. 
It has to be the Primary (RW) or at least Readable Secondary...

That error will only present itself if you have an AG, and applications/users are trying to run update/delete/inserts to the secondary instance/replica DB. 
Trace where those queries are coming from,...

https://docs.microsoft.com/en-us/sql/database-engine/availability-groups/windows/creation-and-configuration-of-availability-groups-sql-server
https://www.brentozar.com/sql/sql-server-alwayson-availability-groups/

Have you tried the -C options with RAW?

-C { ACP | OEM | RAW | code_page }

Did you restart SQL after sp_drop and sp_addserver? You need to. 
Be sure to perform the SP steps:

sp_dropserver <old_name>;GO
sp_addserver <new_name>, local;GO

When checking...

No, but you will lose the option to backup Transaction logs and restore from thereon.
Recovering might end up in data loss.

Provide the table DDL and insert script for some sample data please.

Please post DDL and sample data for quicker assistance.

I've seen 'dbo.SORT' being full specifically when you have set to sort an index rebuild to SORT_IN_TEMPDB.

From your post I assume you have some layer of virtualization, and all VMs running on this one host. 
If you have an EA, you should fully license all the...

Have you considered data compression?